2,450 research outputs found
Uniqueness thresholds on trees versus graphs
Counter to the general notion that the regular tree is the worst case for
decay of correlation between sets and nodes, we produce an example of a
multi-spin interacting system which has uniqueness on the -regular tree but
does not have uniqueness on some infinite -regular graphs.Comment: Published in at http://dx.doi.org/10.1214/07-AAP508 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Rapid Mixing for Lattice Colorings with Fewer Colors
We provide an optimally mixing Markov chain for 6-colorings of the square
lattice on rectangular regions with free, fixed, or toroidal boundary
conditions. This implies that the uniform distribution on the set of such
colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet
has a finite correlation length and a unique Gibbs measure at zero temperature.
Four and five are now the only remaining values of q for which it is not known
whether there exists a rapidly mixing Markov chain for q-colorings of the
square lattice.Comment: Appeared in Proc. LATIN 2004, to appear in JSTA
Coupling with the stationary distribution and improved sampling for colorings and independent sets
We present an improved coupling technique for analyzing the mixing time of
Markov chains. Using our technique, we simplify and extend previous results for
sampling colorings and independent sets. Our approach uses properties of the
stationary distribution to avoid worst-case configurations which arise in the
traditional approach. As an application, we show that for ,
the Glauber dynamics on -colorings of a graph on vertices with maximum
degree converges in steps, assuming and that the graph is triangle-free. Previously, girth was needed.
As a second application, we give a polynomial-time algorithm for sampling
weighted independent sets from the Gibbs distribution of the hard-core lattice
gas model at fugacity , on a regular graph
on vertices of degree and girth . The best
known algorithm for general graphs currently assumes .Comment: Published at http://dx.doi.org/10.1214/105051606000000330 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Rapid Mixing of Gibbs Sampling on Graphs that are Sparse on Average
In this work we show that for every and the Ising model defined
on , there exists a , such that for all with probability going to 1 as , the mixing time of the
dynamics on is polynomial in . Our results are the first
polynomial time mixing results proven for a natural model on for where the parameters of the model do not depend on . They also provide
a rare example where one can prove a polynomial time mixing of Gibbs sampler in
a situation where the actual mixing time is slower than n \polylog(n). Our
proof exploits in novel ways the local treelike structure of Erd\H{o}s-R\'enyi
random graphs, comparison and block dynamics arguments and a recent result of
Weitz.
Our results extend to much more general families of graphs which are sparse
in some average sense and to much more general interactions. In particular,
they apply to any graph for which every vertex of the graph has a
neighborhood of radius in which the induced sub-graph is a
tree union at most edges and where for each simple path in
the sum of the vertex degrees along the path is . Moreover, our
result apply also in the case of arbitrary external fields and provide the
first FPRAS for sampling the Ising distribution in this case. We finally
present a non Markov Chain algorithm for sampling the distribution which is
effective for a wider range of parameters. In particular, for it
applies for all external fields and , where is the critical point for decay of correlation for the Ising model on
.Comment: Corrected proof of Lemma 2.
Spatial Mixing of Coloring Random Graphs
We study the strong spatial mixing (decay of correlation) property of proper
-colorings of random graph with a fixed . The strong spatial
mixing of coloring and related models have been extensively studied on graphs
with bounded maximum degree. However, for typical classes of graphs with
bounded average degree, such as , an easy counterexample shows that
colorings do not exhibit strong spatial mixing with high probability.
Nevertheless, we show that for with and
sufficiently large , with high probability proper -colorings of
random graph exhibit strong spatial mixing with respect to an
arbitrarily fixed vertex. This is the first strong spatial mixing result for
colorings of graphs with unbounded maximum degree. Our analysis of strong
spatial mixing establishes a block-wise correlation decay instead of the
standard point-wise decay, which may be of interest by itself, especially for
graphs with unbounded degree
Correlation decay and partition function zeros: Algorithms and phase transitions
We explore connections between the phenomenon of correlation decay and the
location of Lee-Yang and Fisher zeros for various spin systems. In particular
we show that, in many instances, proofs showing that weak spatial mixing on the
Bethe lattice (infinite -regular tree) implies strong spatial mixing on
all graphs of maximum degree can be lifted to the complex plane,
establishing the absence of zeros of the associated partition function in a
complex neighborhood of the region in parameter space corresponding to strong
spatial mixing. This allows us to give unified proofs of several recent results
of this kind, including the resolution by Peters and Regts of the Sokal
conjecture for the partition function of the hard core lattice gas. It also
allows us to prove new results on the location of Lee-Yang zeros of the
anti-ferromagnetic Ising model.
We show further that our methods extend to the case when weak spatial mixing
on the Bethe lattice is not known to be equivalent to strong spatial mixing on
all graphs. In particular, we show that results on strong spatial mixing in the
anti-ferromagnetic Potts model can be lifted to the complex plane to give new
zero-freeness results for the associated partition function. This extension
allows us to give the first deterministic FPTAS for counting the number of
-colorings of a graph of maximum degree provided only that . This matches the natural bound for randomized algorithms obtained by
a straightforward application of Markov chain Monte Carlo. We also give an
improved version of this result for triangle-free graphs
Strong spatial mixing of list coloring of graphs
The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise known to be #P hard. In the context of coloring, strong spatial mixing has been established for Kelly trees in (Ge and Stefankovic, arXiv:1102.2886v3 (2011)) when q ≥ α[superscript *] Δ + 1 where q the number of colors, Δ is the degree and α[superscript *] is the unique solution to xe[superscript -1/x] = 1. It has also been established in (Goldberg et al., SICOMP 35 (2005) 486–517) for bounded degree lattice graphs whenever q ≥ α[superscript *] Δ - β for some constant β, where Δ is the maximum vertex degree of the graph. We establish strong spatial mixing for a more general problem, namely list coloring, for arbitrary bounded degree triangle-free graphs. Our results hold for any α > α[superscript *] whenever the size of the list of each vertex v is at least αΔ(v) + β where Δ(v) is the degree of vertex v and β is a constant that only depends on α. The result is obtained by proving the decay of correlations of marginal probabilities associated with graph nodes measured using a suitably chosen error function
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